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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 16
MAXWELL AND ELECTROMAGNETISM
James Clerk Maxwell
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Cavendish
Laboratory
Images removed due to copyright restrictions.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Cavendish Laboratory
Images removed due to copyright restrictions.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Last time
1.
2.
3.
4.
p-n junctions, built-in voltage, rectification
Bl h oscillations,
Bloch
ill ti
conductivity
d ti it iin semiconductors
i d t
Electron transport at the nanoscale
Phonons, vibrational free energy, and the quasi-harmonic
approximation
5. Electron-phonon interactions, and phonon-phonon decays
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Study
• Fox,
Fox Optical Properties of Solids
Solids,
Appendix A and Chap 1.
• Prof Fink lecture notes (to be posted)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Image courtesy NASA.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Electric field, polarization,
displacement
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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A complex crystal lattice can have a permanent intrinsic polarization P.
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Polarization in lead titanate
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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A complex crystal lattice can have a permanent intrinsic polarization P.
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A complex crystal lattice can have a permanent intrinsic polarization P.
Figures by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Dielectric constant, susceptibility
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Magnetic response
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Maxwell equations
r
r r 1 ∂B
∇× E +
= 0
c ∂t
r
r r 1 ∂D 4π r
J
∇× H −
=
c ∂t
c
r r
∇⋅B = 0
r r
∇ ⋅ D = 4πρ
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Vector potential and gauges
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Vector potential and gauges
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Summary
r
r r 1 ∂B
∇× E +
=0
c ∂t
r
r r 1 ∂D 4π r
J
∇× H −
=
c ∂t
c
r r
∇⋅B = 0
r r
∇ ⋅ D = 4πρ
r
D=
r
B=
ε{
r
E
dielectric tensor
µ
{
r
H
permeability tensor
r
r r
r
D = ε E = E + 4π P
r
r r
r
B = µ H = H + 4π M
E – electric field
12 variables
H – magnetic field
8 scalar Maxwell
equations
D – electric displacement
B – magnetic displacement
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Electromagnetic waves
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Electromagnetic waves
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Summary
r
r
1
r
⋅
r r 1 ∂B
r ⎛ 1 r r⎞ 1 ∂ r r
1 r r 1 ∂H
⋅∇×
µ
∇× E +
= 0 ⎯⎯
→ ∇× E +
= 0 ⎯⎯
→∇×⎜ ∇× E ⎟ +
∇× H = 0
c ∂t
c ∂t
µ
⎝µ
⎠ c ∂t
r
r
∂
r r 1 ∂D
⋅
1 ∂ r r ε ∂ 2
E
∂t
∇× H −
= 0 ⎯⎯→
∇× H = 2 2
c ∂t
c ∂t
c ∂t
r
r ⎛ 1 r r ⎞ ε ∂2E
∇×⎜ ∇× E ⎟ + 2 2 = 0
⎝µ
⎠ c ∂t
r
2
r 2 r µε ∂ E
∇ E− 2
=0
c ∂t 2
r
r r µε ∂ 2 H
∇2 H − 2
=0
c ∂t 2
r
r 2 r µε ∂ E
∇ E= 2
c ∂t 2
2
rr
r
r
E ( x, y, z, t ) = E0 eiωt − k ⋅r
c r
k =ω
µε
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Refractive index
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Phase velocity
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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Wave packets
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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